Physics 195 Spring 2008 Homework #5

1) 1) Draw several pictures to show what happens as these waves pass through each other. Assume that sharp corners are possible in a wave and use them in your drawings.

a)

b)

2) A student measures the period of a pendulum three times and gets the answers 1.6s, 1.8s, 1.7s.

a) What are the mean and standard deviation?

b) If the student decides to make a fourth measurement, what is the probability that this new measurement will fall outside the range 1.6s to 1.8s?

3) a) Show that f(x,t) = 0.1sin[2π(0.5x - 20t)] satisfies the wave equation

= 1600.

b) What is the propagation velocity?

4) A mechanical oscillator connected to the end of a stretched string creates a transverse displacement of the end of the string given by

ξ = 0.01sin(20t) where ξ is in metres. If the tension in the string is 10 N and the string has mass density 20 g/m, find

a) the velocity of the transverse waves

b) the frequency, f, of the waves, and

c) the wavelength.

5) A nylon guitar string has a linear density of 7.20 g/m and is under a tension of 150 N. The fixed supports are a distance 90.0 cm apart and string oscillates in its third mode.

a) Find the speed of waves on the string.

b) Find the frequency of the normal mode.

c) Find the wavelength of the mode.

6) A string of length L = 1.20m with linear density μ = 1.6g/m is driven at one end by a sinusoidal oscillator with frequency f = 120Hz. The amplitude of the oscillation is small enough that the end of the string can be considered as fixed. The other end goes over a pulley (the 1.20 m is between the driver and the pulley) and is attached to a mass m that hangs down.

a) What mass is needed for the string to oscillate in its fourth mode (n = 4)?

b) What mode can be set up if the mass m = 1.50kg?

7) Two guitar strings have the same length and mass per unit length. The tighter string has a fundamental frequency of 440 Hz. When the two strings are played together you hear beats at 2 Hz. What is the fundamental frequency of the looser string?

8) Find the Fourier series representation for a wave on a string of length L started from rest so that,

y(x,0) = .

10) Two strings are attached at x = 0. String one has mass per unit length μ₁, and string two has mass per unit length μ₂. An incident wave, Y ₀ = A₀ cos(k₁x - ω₁t), is incident on the boundary from the left. The resulting reflected wave is Y _R = A_R cos(-k₁x - ω₁t) and the transmitted wave is Y _T = A_T cos(k₂x - ω₂t).

a) Briefly describe why ω₂ = ω₁ = ω.

b) Describe the two boundary conditions at x = 0 and use them to find two equations relating A_0,A_R, and A_T.

c) Solve these equations to show that

A_R = A₀ and A_T = A₀.

10) If a set of N measurements, x_i, has mean = ∑ _i=1^Nx_i then the deviation of x_i is defined to be d_i = x_i -. Prove that the mean of the deviations is always zero.